The projection of the line segment joining the points $(- 1, 0, 3)$ and $(2, 5, 1)$ on the line whose direction ratios are $(6, 2, 3)$ is

Direction cosines of the line are 

$\begin{array}{l}(\frac{6}{\sqrt{\left\{(6{)}^2+(2{)}^2+(3{)}^2\right\}}},\frac{2}{\sqrt{\left\{(6{)}^2+(2{)}^2+(3{)}^2\right\}}}\\ \frac{3}{\sqrt{\left\{(6{)}^2+(2{)}^2+(3{)}^2\right\}}})\text{ i.e., (}\frac{6}{7},\frac{2}{7},\frac{3}{7})\end{array}$

$\therefore$ Projection of the line segment joining the points on the given line=  $\begin{array}{r}\frac{6}{7}(2+1)+\frac{2}{7}(5-0)+\frac{3}{7}(1-3)=\frac{22}{7}\\ \left[l\left(x_2-x_1\right)+m\left(y_2-y_1\right)+n\left(z_2-z_1\right)\right]\end{array}$